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Holt McDougal Geometry 7-2 Similarity and Transformations 7-2 Similarity and Transformations Holt Geometry Warm Up Lesson Presentation Lesson Quiz Holt McDougal Geometry

# Similarity and Transformations - Neshaminy School District · 2016-03-08 · Holt McDougal Geometry 7-2 Similarity and Transformations Lesson Quiz : Part-III 4. Prove that circle

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• Holt McDougal Geometry

7-2 Similarity and Transformations 7-2 Similarity and Transformations

Holt Geometry

Warm Up

Lesson Presentation

Lesson Quiz

Holt McDougal Geometry

• Holt McDougal Geometry

7-2 Similarity and Transformations

Warm Up

Find the image point when the indicated transformation is applied to the given pre-image point.

1. (x, y) → (x + 3, y – 1); (2, 4) (5, 3)

2. (x, y) → (x, –y); (–2, 1) (-2,-1)

3. (x, y) → (3x, 3y); (–5, 0) (-15,-0)

(1,-2) 4. (x, y) → ; (3, -6) 3 1 x,

3 1 y

• Holt McDougal Geometry

7-2 Similarity and Transformations

Draw and describe similarity transformations in the coordinate plane.

Use properties of similarity transformations to determine whether polygons are similar and to prove circles similar.

Objectives

• Holt McDougal Geometry

7-2 Similarity and Transformations

similarity transformation

Vocabulary

• Holt McDougal Geometry

7-2 Similarity and Transformations

A transformation that produces similar figures is a similarity transformation. A similarity transformation is a dilation or a composite of one or more dilations and one or more congruence transformations. Two figures are similar if and only if there is a similarity transformation that maps one figure to the other figure.

• Holt McDougal Geometry

7-2 Similarity and Transformations

Translations, reflections, and rotations are congruence transformations.

Remember!

• Holt McDougal Geometry

7-2 Similarity and Transformations

Example 1: Drawing and Describing Dilations

D: (x, y) → (3x, 3y)

A(1, 1), B(3, 1), C(3, 2)

dilation with center (0, 0) and scale factor 3

A. Apply the dilation D to the polygon with the given vertices. Describe the dilation.

• Holt McDougal Geometry

7-2 Similarity and Transformations

Example 1: Continued

dilation with center (0, 0) and scale factor 3 4

D: (x, y) →

P(–8, 4), Q(–4, 8), R(4, 4)

4 3 x,

4 3 y

B. Apply the dilation D to the polygon with the given vertices. Describe the dilation.

• Holt McDougal Geometry

7-2 Similarity and Transformations

Check It Out! Example 1

D'(-2, 0), E'(-2, -1), F'(-1, -2); dilation with center

(0, 0) and scale factor 1 4

Apply the dilation D : (x, y)→

polygon with vertices D(-8, 0), E(-8, -4), and F(-4, -8). Name the coordinates of the image points. Describe the dilation.

to the 4 1 x,

4 1 y

• Holt McDougal Geometry

7-2 Similarity and Transformations

A. A(–6, -6), B(-6, 3), C(3, 3),

D(3, -6) and H(-2, -2),

J(-2, 1), K(1, 1), L(1, -2)

Example 2 : Determining Whether Polygons are

Similar

Yes; ABCD maps to HJKL by a dilation:

(x, y) → 1

3 x

1

3 y ,

Determine whether the polygons with the given vertices are similar.

• Holt McDougal Geometry

7-2 Similarity and Transformations

B. P(2, 0), Q(2, 4), R(4, 4),

S(4, 0) and W(5, 0),

X(5, 10), Y(8, 10), Z(8, 0).

Example 2: Continued

No; (x, y) → (2.5x, 2.5y) maps P to W, but not S to Z.

• Holt McDougal Geometry

7-2 Similarity and Transformations

Example 2: Continued

C. A(1, 2), B(2, 2), C(1, 4) and

D(4, -6), E(6, -6), F(4, -2)

Yes; ABC maps to A’B’C’ by a translation: (x, y) → (x + 1, y - 5). Then A’B’C’ maps to DEF by a dilation: (x, y) → (2x, 2y).

• Holt McDougal Geometry

7-2 Similarity and Transformations

D. F(3, 3), G(3, 6), H(9, 3),

J(9, –3) and S(–1, 1),

T(–1, 2), U(–3, 1), V(–3, –1).

Example 2: Continued

Yes; FGHJ maps to F’G’H’J’ by a reflection : (x, y) → (-x, y). Then F’G’H’J’ maps to STUV by a

dilation:

(x, y) 1

3 x

1

3 y ,

• Holt McDougal Geometry

7-2 Similarity and Transformations

Determine whether the polygons with the given vertices are similar : A(2, -1), B(3, -1), C(3, -4) and P(3, 6), Q(3, 9), R(12, 9).

Check It Out! Example 2

The triangles are similar because ABC can be mapped to A'B'C' by a rotation: (x, y) → (-y, x),

and then A'B'C' can be mapped to PQR by a dilation: (x, y) → (3x, 3y).

• Holt McDougal Geometry

7-2 Similarity and Transformations

Example 3: Proving Circles Similar

A. Circle A with center (0, 0) and radius 1 is similar to circle B with center (0, 6) and radius 3.

• Holt McDougal Geometry

7-2 Similarity and Transformations

Example 3: Continued

Circle A can be mapped to circle A’ by a translation: (x, y) → (x, y + 6). Circle A’ and circle

B both have center (0, 6). Then circle A’ can be mapped to circle B by a dilation with center (0, 6) and scale factor 3. So circle A and circle B are similar.

B. Circle C with center (0, –3) and radius 2 is similar to circle D with center (5, 1) and radius 5.

• Holt McDougal Geometry

7-2 Similarity and Transformations

Example 3: Continued

Circle C can be mapped to circle C’ by a translation: (x, y) → (x + 5, y + 4). Circle C’ and

circle D both have center (5, 1).Then circle C’ can be mapped to circle D by a dilation with center (5, 1) and scale factor 2.5. So circle C and circle D are similar.

• Holt McDougal Geometry

7-2 Similarity and Transformations

Prove that circle A with center (2, 1) and radius 4 is similar to circle B with center (-1, -1) and radius 2.

Check It Out! Example 3

Circle A can be mapped to circle A' by a

translation: (x, y) → (x - 3, y - 2). Then circle A'

can be mapped to circle B by a dilation with center

(-1, -1) and scale factor So, circles A and B are

similar.

1 2

• Holt McDougal Geometry

7-2 Similarity and Transformations

Eric wants to make a drawing to show how he will arrange three rugs in his room.

The large rug is twice the size of each small rug. Eric will first draw the small rug in the lower left corner and then the large rug in the middle. How can he draw those rugs?

Example 4: Application

• Holt McDougal Geometry

7-2 Similarity and Transformations

Place the lower left rug in the first quadrant of a coordinate plane in a convenient position. Apply the dilation with center (0, 0) and scale factor 2: (x, y) → (2x, 2y).

Example 4: Continue

• Holt McDougal Geometry

7-2 Similarity and Transformations

What if…? How could Tia draw the middle flag to make it 4 times the size of each of the other flags?

Check It Out! Example 4

Apply the dilation with center (0, 0) and scale

factor 4: (x, y) → (4x, 4y).

• Holt McDougal Geometry

7-2 Similarity and Transformations

Lesson Quiz : Part-I

1. Apply the dilation D: (x, y) to the

polygon with vertices A(2, 4), B(2, 6), and C(6, 4). Name the coordinates of the image points. Describe the dilation.

A’(3, 6), B’(3, 9), C’(9, 6); dilation with center

(0, 0) and scale factor 3

2

• Holt McDougal Geometry

7-2 Similarity and Transformations

Lesson Quiz : Part-II

No; (x, y) → (0.5x, 0.5y) maps A to P, but not B to Q.

Determine whether the polygons with the given vertices are similar.

2. A(-4, 4), B(6, 4), C(6, -4),

D(-4, -4) and P(-2, 2),

Q(4, 2), R(4, -2), S(-2, -2)

3. A(2, 2), B(2, 4), C(6, 4) and

D(3, -3), E(3, -6), F(9, -6)

Yes; △ ABC maps to △ A’B’C’ by a reflection: (x, y) → (x, -y). Then △ A’B’C’ maps to △DEF by a dilation:(x, y) → (1.5x, 1.5y).

• Holt McDougal Geometry

7-2 Similarity and Transformations

Lesson Quiz : Part-III

4. Prove that circle A with center (0, 4) and radius 4 is similar to circle B with center (-2, -7) and radius 6.

Circle A can be mapped to circle A’ by a translation:

(x, y) → (x - 2, y - 11). Circle A’ and circle B both

have center (-2, -7). Then circle A’ can be mapped to

circle B by a dilation with center (-2, -7) and scale

factor 3 2

So circle A and circle B are similar.

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